What digits is the "number" $\infty$ composed of?
You are confusing numbers with numerals. Numerals are symbols that represent numbers. Numbers do not have any intrinsic representation as sequences of digits or anything else. Instead, we devise different schemes for representing numbers with numerals. For example, in one scheme, we use sequences of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent certain numbers; the numeral 119 represents a certain number. But there is nothing privileged or special about this numeral; in a different, similar system, the same number is represented with the numeral 1110111; in a different, less similar system the same number is represented with the numeral 百十九, in another system it is represented with the numeral CXIX, in a different system it is represented with the numeral one hundred and nineteen, and in a different system again it is represented with a certain pattern of electron flow in a chunk of silicon.
So the question of whether a certain number "has digits in it" is a category error. Numbers never have digits. Some systems of numeration use digits, and numerals in those systems have digits in them. But the number of digits will depend on which system you are using. 119 is a three digit numeral, and 1110111 is a seven-digit numeral, but they both represent the same number.
The question that does make sense to ask is whether a certain system of numerals can represent a certain number. For example, some systems are able to represent the number one-half. One might write it in one system as $\frac12$, and in another system as 0.5. Some systems simply have no representation for one-half.
So we can ask if the standard decimal system, the one which uses digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, has a representation of the number infinity, and if so how many digits are used to represent it. And the answer is no, as usually understood, this system has no representation for the number infinity. (Or, more precisely, for any of the several numbers called "infinity".)
There is no ambiguity regarding infinity, particularly with regard to whether or not it is a number. There is, perhaps, some ambiguity to the novice reader regarding infinity, as the term infinity pops up in a lot of different places in mathematics, and can refer to a number of distinct (though related) concepts. For example:
In analysis, we are often interested in describing the limiting behaviour of sequences. Some sequences oscillate, such as the sequence $$1, -1, 1, -1, 1, -1, \dotsc = \{ (-1)^n \}_{n\in\mathbb{N}};$$ some sequences "approach a limit", such as $$ 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dotsc = \left\{ \frac{1}{n} \right\}_{n\in\mathbb{N}}$$ which "approaches" zero; and some sequences "blow up" or are "unbounded", such as $$ 1, 2, 3, 4, 5, \dotsc = \{n\}_{n\in\mathbb{N}}.$$ In the case of an unbounded sequence, we use the shorthand $$ \lim_{n\to\infty} a_n = \infty. $$ If we want to be very formal about this, what it means is that if we pick any number $M$ (no matter how large), we can always find some natural number $N$ so that $a_n$ is bigger than $M$ whenever $n$ is bigger than $N$. In other words, this notation means that the terms $a_n$ grow without bound. To avoid having to write down a bunch of technical notation every time we see something like this, we give a single definition of what it means for "a limit to be equal to infinity," then use the shorthand notation shown above. The symbol $-\infty$ ("negative infinity") is understood similarly.
The overall point is that $-\infty$ and $+\infty$ are not a real numbers in this setting. They can be seen as useful symbols which give a shorthand for a more technical definition, or they may be regarded as points in a topological space which extends the real numbers (which would make $\pm\infty$ affinely-extended real numbers). They don't behave like real numbers, as, for example, $0\cdot\infty$ and $\infty-\infty$ cannot be sensibly defined.
In set theory, it is often desirable to describe the cardinality of a set, where cardinality is a more rigorous way of describing "size". In a way that can be made formal, there is no "largest" natural number, so the set of natural numbers must be of infinite size (i.e. it has infinite cardinality). There are infinitely many natural numbers—here, the use of "infinity" is somewhat distinct from the application to analysis. "Infinity" means something slightly different here. However, there is no ambiguity—the context of the usage makes it clear what is going on. That being said, there is also notation for this kind of infinity: the cardinality of the natural numbers is often denoted by $$ \operatorname{card}(\mathbb{N}) = \aleph_0, $$ where that last symbol is the Hebrew letter "aleph". Something fun to know is that in this context, there are different of infinities. For example, $$ \operatorname{card}(\mathbb{R}) = \mathfrak{c}, $$ where, in a way that can be made formal, $\aleph_0 < \mathfrak{c}$.
Note that $\aleph_0$, $\mathfrak{c}$, and their compatriots (there are other similar symbols with related meanings) are not numbers, though each one can reasonably called "infinity." In the language of mathematics, these guys are "infinite cardinals."
There is also a notion of an "infinite ordinal." For example, the symbol $\omega$ (a lowercase Greek omega) is is a number-like object which has the property that $$ x < \omega $$ for all real numbers $x$. In some sense, $\omega$ is probably the closest thing to a "number" infinity that I am going to discuss. We can do arithmetic with $\omega$, for example (though that arithmetic is a little more complicated than the usual—for example $1+\omega \ne \omega + 1$), but $\omega$ is still not really a number. Alternatively, if you want to call $\omega$ a number, you have to be very careful about defining what you mean by "number" before you go any further. In particular, $\omega$ doesn't fit in well with the real numbers. Instead, $\omega$ is useful in providing a generalization of the natural numbers.
In particular, the ordinals extend the idea of indexing. Given a set, we may want to label every element of that set so that the elements can be put into order. This kind of thing is useful when arguing by induction, for example. The "natural" index set is $\mathbb{N}$, but what if we need indices beyond those contained in $\mathbb{N}$? The symbol $\omega$ is simply if first index after $\mathbb{N}$, and $\omega+1$ is the next index after that. Infinite ordinals extend the idea of addressing objects by a well-order (i.e. of saying what comes first, second, next, &c.).
Note that in each of these contexts, the notion of "infinity" is unambiguous, at least from a mathematical point of view. The notion of "number" is also unambiguous here, though, again, this notion might depend on context. For example, we might desire limits to exist in our space, so we might define $\pm\infty$ to be numbers. This gives us the set of extended real numbers, i.e. $$ \overline{\mathbb{R}} := \mathbb{R} \cup \{-\infty,+\infty\}. $$ Here, positive and negative infinity are numbers, but they are not real numbers—they are extended real numbers. Something similar can probably be done in other contexts, as well.
To address your second question: when you describe the "digits" of a number, you are talking about a decimal expansion or representation of a real number (where we include the natural numbers, integers, and rationals as subsets of the reals). If an object is not a real number, then it does not make any sense to discuss its digits. Infinity is not a real number. Conclude from this what you will.
To adress the "digits" part of the question, while sticking to decimal representation, let's begin by defining what it really means for a number to have certain digits.
For exmple, "one hundred and twenty three point four" has the decimal expression $123.4$ because $123.4 = 1\cdot \textbf{100} + 2 \cdot \textbf{10} + 3\cdot \textbf{1} + 4\cdot \textbf{0.1}$.
Similarly 1/3 can be written as $$3 \cdot \textbf{0.1} + 3 \cdot \textbf{0.01} + 3 \cdot \textbf{0.001} + 3 \cdot \textbf{0.0001} + ... = \sum_{k=1}^{\infty} 3 \cdot 10^{-k}$$
For simplicity, let's forget the decimals parts and focus on integers. A number $n$ will have a decimal expression $a_{k} a_{k-1}... a_1 a_0$ if, and only if, it can be expressed as:
$$\sum_{i=0}^{k} a_i \cdot 10^i$$
Note that this only makes sense for number with a finite ($k$) number of digits, so does not really apply to the case of "infinity". But still, similarly to how we can get decimal expressions for numbers like $\frac{1}{3}$ using infinitely many numbers after the decimal point, one could ask himself whether it is also posible to work with infinitely many numbers before the decimal point. Writing something like:
$$\sum_{i=0}^{\infty} a_i \cdot 10^i$$
The problem is that the above sum will diverge to infinity unless there is a number $k$ such that $a_i=0$ for every $i>k$. In other words, with these "decimal expressions", you can get to infinity in as many ways as you want. In short, if we accepted that convention $33333333....=\infty$, but also $44444444....=\infty$, and $123456789123456789....=\infty$
In conclusion, there is no "standard" way of writing numbers that assigns a particular set of digits to "infinity". If we want to expand our "common-use" decimal numbers to have such a feature, we will have to accept that "infinity" can be written pretty much anyway we like, which does not have a lot of practical use.